A235488 Squarefree numbers which yield zero when their prime factors are xored together.

70, 646, 1798, 2145, 3526, 5865, 6006, 9177, 11305, 13110, 16422, 20553, 20806, 21489, 23529, 28905, 28985, 30305, 31465, 37961, 38086, 38454, 42441, 44022, 44998, 45353, 45942, 46345, 53985, 54230, 55913, 60630, 60697, 61705, 62049, 64790, 78406, 80934, 81158

Offset: 1

Author

Antti Karttunen, Jan 22 2014

Keywords

• nonn
• base

Comments

All n for which A008683(n) <> 0 and A072594(n) = 0.

It seems that an analogous case as A072595 for GF(2)[X]-polynomials is just the squares of GF(2)[X]-polynomials (A000695), thus in that ring, the sequence analogous to this one would be empty.

This sequence happens also to encode in the prime factorization of n a certain subset of the Nim game positions that are second-player win.

Examples

			70 is included, as 70 = 2*5*7, whose binary representations are '10', '101' and '111', which when all are xored (cf. A003987) together, cancel all 1-bits, thus yielding zero.
212585 is included, as 212585 = 5*17*41*61, and when we xor their base-2 representations together:
     101
   10001
  101001
  111101
--------
  000000
we get only zeros, because in each column (bit-position), there is an even number of 1-bits.
		

Links

• Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 75 terms from Karttunen)
• Wikipedia, Nim

Crossrefs

Intersection of A005117 and A072595 (equally: of A005117 and A072596).

Cf. A003987, A235490, A079599, A048833, A050314.

Programs

Mathematica

Select[Range[82000],SquareFreeQ[#]&&BitXor@@FactorInteger[#][[All,1]]==0&] (* Harvey P. Dale, Apr 01 2017 *)

PARI

is(n)=if(n<9, return(0)); my(f=factor(n)); vecmax(f[,2])==1 && fold(bitxor, f[,1])==0 \\ Charles R Greathouse IV, Aug 06 2016